Question

Find the quotient. $$ \frac{2 x^2-12 x}{x^2-7 x+6} \div \frac{2 x}{3 x-3} $$

Short answer

The quotient is \( \frac{3(x-6)(x-1)}{(x-2)(x-3)} \)

Step-by-step solution

Simplify The Expressions

First, simplify both fractions as much as possible by factoring. The first expression becomes \( \frac{2 x(x-6)}{x^2-7x+6} \) and the second one simplifies to \( \frac{2x}{3(x-1)} \).

Factorize the denominator

Next, factorize the denominator of the first expression. The expression will be: \( \frac{2x(x-6)}{(x-2)(x-3)} \)

Convert Division Into Multiplication

Now, change the operation from division to multiplication by using the reciprocal of the divisor, it becomes a multiplication problem: \( \frac{2x(x-6)}{(x-2)(x-3)} \cdot \frac{3(x-1)}{2x} \)

Simplify the new expression

Now, simplify the new fractional expression by cancelling the terms that appear on both the numerator and the denominator. The solution is: \( \frac{3(x-6)(x-1)}{(x-2)(x-3)} \)

Key concepts

Simplifying Algebraic Fractions

Simplifying algebraic fractions is an essential skill in algebra that involves reducing fractions to their simplest form. This often requires factoring polynomials in both the numerator and denominator and then canceling out common factors. Consider the algebraic fraction \(\frac{2 x^2-12 x}{x^2-7 x+6}\). To simplify, you would first look for common factors in the numerator, which in this case is 2x. Then you would factor the denominator into \(x^2-7x+6 = (x-2)(x-3)\), revealing the factors that make up the polynomial.

Once the fraction is fully factored, you can easily identify and cancel out any common terms in the numerator and the denominator. This step is crucial as it simplifies the expression, making it easier to handle in subsequent calculations, particularly when performing operations like addition, subtraction, multiplication, or division with other fractions.

Factoring Polynomials

Factoring polynomials is a method used to break down a polynomial into the product of simpler polynomials. When looking at a polynomial like \(x^2-7x+6\), we search for two numbers that multiply to give the constant term (in this case, 6) and add up to the coefficient of the x-term (here, -7).

For \(x^2-7x+6\), the numbers are -2 and -3. So, this polynomial factors as \( (x-2)(x-3) \). Factoring is an invaluable tool when working with algebraic fractions. It allows us to simplify expressions, solve quadratic equations, and perform operations with algebraic fractions more easily. Always look for the greatest common factor or patterns such as the difference of squares or perfect square trinomials to factor polynomials effectively.

Reciprocal of a Fraction

The reciprocal of a fraction is simply a flipped version of the original fraction, where the numerator becomes the denominator and vice versa. For instance, the reciprocal of \(\frac{2x}{3(x-1)}\) is \(\frac{3(x-1)}{2x}\).

Understanding the concept of a reciprocal is very important when dividing one fraction by another. To divide fractions, you multiply the first fraction by the reciprocal of the second. This is why, in the given exercise, the division of \(\frac{2 x^2-12 x}{x^2-7 x+6} \div \frac{2 x}{3 x-3}\) is converted into a multiplication problem. It's a fundamental process that simplifies the division of fractions into a more familiar operation.

Multiplication of Rational Expressions

Multiplication of rational expressions follows the same basic principles of multiplying numerical fractions. Simply multiply the numerators together and multiply the denominators together. However, when working with algebraic expressions, there can be opportunities to simplify before multiplying.

By factoring and cutting down the expressions first, as done with the expression \(\frac{2x(x-6)}{(x-2)(x-3)} \cdot \frac{3(x-1)}{2x}\) in the given exercise, you can cancel out any common terms, thereby simplifying the multiplication process. This simplification step is not just about making the numbers smaller; it can also reveal cancellation opportunities that wouldn't be obvious had multiplication been performed immediately.